Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem

Abstract

We continue the study of compactness phenomena between the set-theoretic universe and $\mathrm{HOD}$ initiated by Goldberg--Poveda \cite{GolPov}. We focus on compactness phenomena around the power-set functions of $V$ and $\mathrm{HOD}$. We prove: (1) A singular strong limit cardinal with uncountable cofinality cannot be the first place where $\mathcal{P}(\cdot )$ and $ \mathcal{P}^{\mathrm{HOD}}(\cdot)$ disagree. (2) Assuming the existence of a measurable cardinal, $\aleph_ω$ can be the first place where $\mathcal{P}(\aleph_ω)\neq \mathcal{P}^{\mathrm{HOD}}(\aleph_ω)$, answering a question of Hayut. (3) If $κ$ is strong limit singular of uncountable cofinality, $\mathrm{HOD}$ is correct about cardinals less than or equal to $κ^+$ and the GCH holds in $\mathrm{HOD}$ below $κ^+$ then $(\mathrm{HOD}, V)$ has the $\mathrm{cf}(κ)^+$-cover property. We also show that the GCH assumption in (3) is necessary, which demonstrates that Magidor's classical Covering Theorem is optimal.